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Cauchy trace for some locally integrable functions on a bounded open subset of \(\mathbb C\). (Trace de Cauchy pour certaines fonctions localement intégrables sur un ouvert borné de \(\mathbb C\).) (French. English summary) Zbl 1122.30026
Summary: Let \(\Omega\) be a bounded open subset of \(\mathbb C\), and let \(f\) be a distribution on \(\Omega\) such that \(\overline{\partial} f\) is a Radon measure of finite total mass. By means of the Cauchy transform, we introduce the “Cauchy trace” of \(f\), which takes values in the set of analytic functionals on the boundary \(\partial \Omega\) of \(\Omega\). The properties of this application are studied in detail. For instance, the characterization of its kernel is discussed according to the properties of the boundary \(\partial \Omega\). Roughly speaking, the Cauchy trace allows us to interpret the Cauchy-Pompeiu formula in the same way as the Sobolev trace allows to interpret the Stokes formula.

MSC:
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
46E15 Banach spaces of continuous, differentiable or analytic functions
46F05 Topological linear spaces of test functions, distributions and ultradistributions
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